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  2. Dirichlet's approximation theorem - Wikipedia

    en.wikipedia.org/wiki/Dirichlet's_approximation...

    In his Essai sur la théorie des nombres (1798), Adrien-Marie Legendre derives a necessary and sufficient condition for a rational number to be a convergent of the simple continued fraction of a given real number. [4] A consequence of this criterion, often called Legendre's theorem within the study of continued fractions, is as follows: [5 ...

  3. Hermite's problem - Wikipedia

    en.wikipedia.org/wiki/Hermite's_problem

    Rational numbers are algebraic numbers that satisfy a polynomial of degree 1, while quadratic irrationals are algebraic numbers that satisfy a polynomial of degree 2. For both these sets of numbers we have a way to construct a sequence of natural numbers (a n) with the property that each sequence gives a unique real number and such that this real number belongs to the corresponding set if and ...

  4. Rational number - Wikipedia

    en.wikipedia.org/wiki/Rational_number

    In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective rational sometimes means that the coefficients are rational numbers. For example, a rational point is a point with rational coordinates (i.e., a point whose coordinates are rational numbers); a rational matrix is a matrix of rational numbers; a rational polynomial may be a polynomial with rational ...

  5. Rhind Mathematical Papyrus 2/n table - Wikipedia

    en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus...

    The table consisted of 26 unit fraction series of the form 1/n written as sums of other rational numbers. [9] The Akhmim wooden tablet wrote difficult fractions of the form 1/n (specifically, 1/3, 1/7, 1/10, 1/11 and 1/13) in terms of Eye of Horus fractions which were fractions of the form ⁠ 1 / 2 k ⁠ and remainders expressed in terms of a ...

  6. Partial fraction decomposition - Wikipedia

    en.wikipedia.org/wiki/Partial_fraction_decomposition

    In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. [1]

  7. Algebraic number theory - Wikipedia

    en.wikipedia.org/wiki/Algebraic_number_theory

    Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers , finite fields , and function fields .

  8. Arithmetic - Wikipedia

    en.wikipedia.org/wiki/Arithmetic

    Rational number arithmetic is the branch of arithmetic that deals with the manipulation of numbers that can be expressed as a ratio of two integers. [93] Most arithmetic operations on rational numbers can be calculated by performing a series of integer arithmetic operations on the numerators and the denominators of the involved numbers.

  9. Calkin–Wilf tree - Wikipedia

    en.wikipedia.org/wiki/Calkin–Wilf_tree

    In number theory, the Calkin–Wilf tree is a tree in which the vertices correspond one-to-one to the positive rational numbers.The tree is rooted at the number 1, and any rational number q expressed in simplest terms as the fraction ⁠ a / b ⁠ has as its two children the numbers ⁠ 1 / 1+1/q ⁠ = ⁠ a / a + b ⁠ and q + 1 = ⁠ a + b / b ⁠.

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